# The Seifert-van Kampen Theorem Example 2

On The Seifert-van Kampen Theorem page we stated the very important Seifert-van Kampen theorem. We will now look at some examples of applying the theorem. More examples can be found on the following pages:

**The Seifert-van Kampen Theorem Example 2**

## Example 2

**Let $X$ be the torus with a line segment that touches the torus only at its endpoints as depicted below. Use the Seifert-van Kampen theorem to find the fundamental group of $X$.**

The line across the hole of the torus can be deformed to:

Let $U_1$ and $U_2$ be the open sets depicted below:

We see that the circle is a deformation retract of $U_1$. We also see that the torus is a deformation retract of $U_2$. Lastly we see that a single point is a deformation retract of $U_1 \cap U_2$. Therefore:

(1)Therefore by the Seifert-van Kampen theorem we have that:

(2)